7.9
Relation Entre la Force de cisaillement et le Moment de flexion
Lorsqu'une poutre est soumise à diverses charges, telles qu'une charge répartie, des charges concentrées et un moment de couple, elle subit à la fois…
Supposons qu’une poutre soit soumise à une charge répartie, à deux charges concentrées et à un moment de couple. Etablir la relation entre le cisaillement et le moment de flexion.
Considérez une section élémentaire de la poutre et dessinez un schéma de corps libre de la section.
Pour que la section soit en équilibre, le moment agissant sur le côté droit de la section doit être plus élevé d’une petite quantité finie.
Une force résultante de la charge répartie est exercée à une distance fractionnaire de l'extrémité droite de la section.
En utilisant l’équation d’équilibre du moment, une relation entre le moment et le cisaillement peut être obtenue.
De plus, en le divisant par Δx et en laissant Δx approcher de zéro, on détermine la pente du diagramme du moment, ce qui équivaut au cisaillement.
Un moment de flexion maximal se produit au point où la pente du moment et le cisaillement sont nuls.
En intégrant la charge répartie sur la section élémentaire, située entre deux points arbitraires, une corrélation entre la variation du moment de flexion et l’aire sous le diagramme de cisaillement est obtenue.
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Q1: How do you establish the relationship between shear and bending moment in a beam?
To establish the relationship, analyze an elemental section of the beam using a free-body diagram. Apply equilibrium equations for moments, noting that the moment on the right side should be higher by a small amount. By dividing the moment equation by Δx and letting Δx approach zero, you determine the slope of the moment diagram, which equals the shear force.
Q2: What does the slope of the moment diagram represent?
The slope of the moment diagram is equivalent to the shear force acting on the beam. This relationship is derived by dividing the moment equilibrium equation by the elemental length Δx and taking the limit as Δx approaches zero, establishing a direct mathematical connection between moment and shear.
Q3: Where does the maximum bending moment occur on a beam?
The maximum bending moment occurs at the point where both the slope of the moment diagram and the shear force are zero. This critical location represents a transition point where the shear changes sign, making it essential for structural design and analysis.
Q4: How does the area under the shear diagram relate to bending moment?
By integrating the distributed load over an elemental section between two arbitrary points, a correlation emerges: the change in bending moment equals the area under the shear diagram. This relationship provides a graphical method for determining moment variations along the beam's length.
Q5: Why must the moment on the right side of an elemental section be higher than on the left?
For the elemental section to remain in equilibrium, the moment on the right side must be higher by a small finite amount to balance the effects of shear forces and distributed loads acting on the section. This moment difference is essential for maintaining rotational equilibrium of the infinitesimal element.
Q6: What role does the distributed load play in the shear-moment relationship?
The distributed load exerts a resultant force at a fractional distance from the section's right end, creating the moment differential needed for equilibrium. When integrated over the elemental section, the distributed load determines how shear and moment vary along the beam, directly influencing the shape of both diagrams.
Q7: How do concentrated loads and couple moments affect the shear-moment relationship?
Concentrated loads and couple moments contribute to the overall loading condition on the beam, affecting both shear and bending moment distributions. These loads must be included in the free-body diagram analysis to accurately establish the relationship between shear and moment throughout the beam.